Page 90 - Matrix Analysis & Applied Linear Algebra

84 Chapter 3 Matrix Algebra

Conjugate Transpose
For A =[a ij ], the conjugate matrix is defined to be A =[a ij ] , and
¯ T
T
the conjugate transpose of A is defined to be A = A . From now
¯ T
on, A will be denoted by A , so [A ] = a ji . For example,
∗
∗
ij
 
∗ 1+4i 3
1 − 4i i 2
=  −i 2 − i  .
3 2+i 0
2 0
∗ ∗
(A ) = A for all matrices, and A = A T whenever A contains only
∗
∗
real entries. Sometimes the matrix A is called the adjoint of A.
The transpose (and conjugate transpose) operation is easily combined with
matrix addition and scalar multiplication. The basic rules are given below.
Properties of the Transpose
If A and B are two matrices of the same shape, and if α is a scalar,
then each of the following statements is true.

T T T ∗
∗
∗
(A + B) = A + B and (A + B) = A + B . (3.2.1)
T T ∗
∗
(αA) = αA and (αA) = αA . (3.2.2)

17
Proof. We will prove that (3.2.1) and (3.2.2) hold for the transpose operation.
The proofs of the statements involving conjugate transposes are similar and are
left as exercises. For each i and j, it is true that
T T T T T
[(A + B) ] ij =[A + B] =[A] ji +[B] ji =[A ] ij +[B ] ij =[A + B ] ij .
ji
17
Computers can outperform people in many respects in that they do arithmetic much faster
and more accurately than we can,and they are now rather adept at symbolic computation and
mechanical manipulation of formulas. But computers can’t do mathematics—people still hold
the monopoly. Mathematics emanates from the uniquely human capacity to reason abstractly
in a creative and logical manner,and learning mathematics goes hand-in-hand with learning
how to reason abstractly and create logical arguments. This is true regardless of whether your
orientation is applied or theoretical. For this reason,formal proofs will appear more frequently
as the text evolves,and it is expected that your level of comprehension as well as your ability
to create proofs will grow as you proceed.